Finally, given a Lie algebra and a representation, we introduce Rota-Baxter paired operators that induces a weighted Rota-Baxter Lie algebra and representation of it. The terminology of Rota-Baxter paired operators is motivated by a paper by Zheng, Guo and Zhang [1]} where the authors introduced Rota-Baxter paired modules in the associative context. We define a suitable differential graded Lie algebra that characterize Rota-Baxter paired operators as its Maurer-Cartan elements. This suggests us to define the cohomology of Rota-Baxter paired operators that control formal deformations.
